Your issue is algorithm
is a float environment by default, and you can't have floats inside a minipage
. However, you can specify the [H]
option to suppress floating and have it work as you intend:
\documentclass{article}
\usepackage{algorithm2e}
\begin{document}
\begin{minipage}{.5\linewidth}
\begin{algorithm}[H]\small
{Start}\;
\eIf{A is bigger than B}{
\eIf{if A is bigger than C}{
A;
}{
C;
}
}{
\eIf{B is bigger than C}{
B;
}{
C;
}
}
\caption{A B C comparison}
\end{algorithm}
\end{minipage}
\end{document}

In the future please include a MWE showing your \documentclass
and \usepackage
commands, helps people help you! I've never used algorythm2e
so it took me a bit to find the right package.